Question

Rolle's theorem is not applicable in which one of the following cases?

• A. $f\left(x\right)=|x|\text{in}[-2,2]$
• B. $f\left(x\right)=x^2-4x+5\text{in}[1,3]$
• C. $f\left(x\right)=\left[x\right]\text{in}[2.5,2.7]$
• D. $f\left(x\right)=x^2-x\text{in}[0,1]$

Answer 1

The correct option is A $f\left(x\right)=|x|\text{in}[-2,2]$

For a function $f:[a,b]\to R$, Rolle's theorem is applicable iff the following three conditions are satisfied:
$\left(i\right)f$ is continuous on $[a,b]$.
$\left(ii\right)f$ is differentiable on $(a,b)$.
$\left(iii\right)f\left(a\right)=f\left(b\right)$.

Option (a) does not satisfy the second condition as $f\left(x\right)=|x|$ is not differentiable at $x=0$.
All other options satisfy the three conditions of Rolle's theorem.